View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 312 (2012) 3467–3472 Contents lists available at SciVerse ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc Indicated coloring of graphs Andrzej Grzesik Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Prof. St. Lojasiewicza 6, PL-30-348 Krakow, Poland article info a b s t r a c t Article history: We study a graph coloring game in which two players collectively color the vertices of a Received 8 March 2011 graph in the following way. In each round the first player (Ann) selects a vertex, and then Accepted 2 July 2012 the second player (Ben) colors it properly, using a fixed set of colors. The goal of Ann is to Available online 26 July 2012 achieve a proper coloring of the whole graph, while Ben is trying to prevent realization of this project. The smallest number of colors necessary for Ann to win the game on a graph G Keywords: (regardless of Ben's strategy) is called the indicated chromatic number of G, and denoted by Graph coloring G . We approach the question how much G differs from the usual chromatic number Chromatic number χi. / χi. / Game chromatic number χ.G/. In particular, whether there is a function f such that χi.G/ 6 f (χ.G// for every graph Random graph G. We prove that f cannot be linear with leading coefficient less than 4=3. On the other Clique-pair conjecture hand, we show that the indicated chromatic number of random graphs is bounded roughly by 4χ.G/. We also exhibit several classes of graphs for which χi.G/ D χ.G/ and show that this equality for any class of perfect graphs implies Clique-Pair Conjecture for this class of graphs. ' 2012 Elsevier B.V. All rights reserved. 1. Introduction Suppose that two players, Ann and Ben, are jointly coloring a graph G using a fixed set of colors C. In each round Ann picks an uncolored vertex and colors it, and then Ben is doing the same. They both agree on respecting the rule of proper coloring: none of them is allowed to create a monochromatic edge. The game stops when either the whole graph is successfully colored, or all colors from C occur on vertices adjacent to some uncolored vertex. In the former case Ann is the winner, in the latter case—Ben. The minimum size of color set C guaranteeing a win for Ann is called the game chromatic number of a graph G, and is denoted by χg .G/. The idea of game coloring in the above form was introduced independently by Bodlaender [4] and Gardner [9]. It was originally motivated by the four color problem and computational complexity issues. Since then the topic developed into several directions leading to deep results, sophisticated methods, and challenging open problems (see a recent survey [3]). There are also some unexpected connections to other areas, such as, for instance, the surprising application of game coloring to graph packing discovered by Kierstead and Kostochka [12]. In this paper we study a variant of the graph coloring game proposed by Grytczuk [10]. In this modification the roles of players are highly asymmetric: in one round Ann is only picking a vertex while Ben is choosing a color for this vertex. All other rules and goals of the players remain the same. So, Ben is not allowed to create a monochromatic edge, but tries to ``block'' some vertex by using all colors from C on its neighbors before Ann will pick it. The minimum number of colors needed for Ann to win this game on a graph G is denoted by χi.G/, and is called the indicated chromatic number of G. At first glance χi.G/ behaves more tamely than χg .G/. Indeed, it is not hard to see that for bipartite graphs we have χi.G/ D 2, while χg .G/ takes arbitrarily large values in this class. Also χi.G/ 6 col.G/ (the coloring number of G, defined precisely in Section5), which is far from the truth for χg .G/. The question we approach is whether the indicated chromatic E-mail address: [email protected]. 0012-365X/$ – see front matter ' 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2012.07.001 3468 A. Grzesik / Discrete Mathematics 312 (2012) 3467–3472 Fig. 1. Twisted diamond graph J for which χi.J/ > χ.J/. number is χ-bounded. More precisely, let I.n/ be the supremum of the values χi.G/ for graphs satisfying χ.G/ 6 n, where χ.G/ denotes the chromatic number of G. We believe that I.n/ is always finite. We show that the indicated chromatic number of the random graph Gn;p is bounded by roughly 4χ.Gn;p/. We also examine a strange Zhu-type question [14] (asked originally for the game chromatic number) whether enlarging the number of colors could be in favor of Ben. We show that there is a graph H such that χi.H/ D 3, but if there are four colors available, then it is much harder for Ann to win the game on H. In the final section we present some connection of the indicated chromatic number of perfect graphs with the Clique-Pair Conjecture stated by Fonlupt and Sebö [7]. 2. Upper bound We start with showing that the difference between χi.G/ and χ.G/ can be arbitrarily big. For this purpose we need a graph G for which χi.G/ is strictly bigger than χ.G/. As χi.G/ D χ.G/ for every bipartite graph G (which is easy to see), we made a search among 3-chromatic graphs. One of the simplest examples is depicted in Fig. 1. We call this graph the twisted diamond and denote it by the letter J. This graph was found by Haªuszczak by computer search. Lemma 1. The twisted diamond graph satisfies χi.J/ D 4. Proof. First notice that J is uniquely 3-colorable. This is so because any proper coloring of the subgraph induced by the vertices ABCD requires that A and D has the same color. Similarly E and H must have the same color. Vertices D and H are adjacent, so their colors should be different. Now, it is easy to see that there is only one way to put colors on the other vertices. We need to prove that there is no winning strategy for Ann when the game is played using three colors. By the unique colorability of J, Ann cannot leave any decision to Ben. Otherwise Ben could choose a wrong color which violates the unique coloring of J, and win the game in consequence. This type of Ann's moves will be called forcing moves. We will prove that such forcing play is impossible for Ann. Let J1 denote the induced subgraph on vertices A, B, C, D and let J2 be the induced subgraph on vertices E, F, G and H. Ann cannot start with two non-adjacent vertices. If Ann starts with two adjacent vertices from one of these subgraphs, she can color this subgraph using forcing moves, but then she cannot do any forcing move to the second of these subgraphs. If Ann starts with two adjacent vertices, but one in J1 and the other in J2, she also cannot do a forcing move, because there are no triangles containing an edge between J1 and J2. This completes the proof. Using this lemma we can prove the aforementioned lower bound on the function I.n/. 4 Theorem 2. The function I.n/ satisfies I.n/ > 3 n for every n divisible by 3. Proof. If we take a graph Jk as k-copies of the twisted diamond graph J connected by all the possible edges, from the above lemma we get χi.Jk/ D 4k for every k > 1. This completes the proof as χ.Jk/ D 3k. This theorem excludes the possibility of a linear upper bound for I.n/ with leading coefficient less than 4=3, but perhaps I.n/ 6 2n. We think that I.n/ is always finite, but we are not aware of a proof of that even for n D 3. On the other hand, no 3-chromatic graph is known with χi.G/ > 5. The hardness of determining the indicated chromatic number may be partially explained by the fact that this number is not monotonic with respect to taking subgraphs. Indeed, the twisted diamond J is a subgraph of the full tripartite graph K3;3;3, which has χi.K3;3;3/ D 3 which is less than χi.J/ D 4. Even if we restrict to induced subgraphs, then a similar obstacle holds. Consider a twisted diamond J with a new vertex I connected to vertices A, B, D, E, G and H. To win with three colors Ann presents vertex I, which gets the first color, and then indicates vertex A, which gets a different color. In subsequent moves Ann can always indicate a vertex which is connected to two vertices having different colors and not connected with a vertex in the third color. It forces all Ben's choices and ends up with good coloring of the whole graph using 3 colors. A. Grzesik / Discrete Mathematics 312 (2012) 3467–3472 3469 3. Random graphs Let Gn;p stand for the probability space of all labeled graphs on n vertices, where every edge appears independently with probability p (see [1,11]).